Bryan Caplan  

The Math Myth

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Math professor David Edwards sent me this short essay on mathematical education.  Printed with his permission.


The Math Myth


The math myth is the myth that the future of the American economy is dependent upon the masses having higher mathematics skills. This myth goes back to at least Sputnik, when the Russians were going to surpass us because they were better in math and science. It returned in the late 80's when the Germans and Japanese were going to surpass us because they were better in math and science. It's occurring again now because the Indians and Chinese are better than us in math and science.
  
I find it difficult to find anyone who uses more than Excel and eighth grade level mathematics (=arithmetic, and a little bit of algebra, statistics and programming). In the summer of 2007 I taught an advanced geometry course and had two students in the class who had been engineers and one who had been an actuary. They claimed never to have used anything beyond Excel and eighth grade level mathematics; never a trig function or even a log or exponential function! There is in fact a deskilling going on in our economy, where even the ability to make change is about to disappear as an important skill.
  
Vivek Wadhwa has described how there's no shortage of scientists and engineers. I've been concerned with what skills those who are working as scientists and engineers actually use. I find that the vast majority of scientists, engineers and actuaries only use Excel and eighth grade level mathematics. This suggests that most jobs that currently require advanced technical degrees are using that requirement simply as a filter. In particular, I'm working on documenting the following:
  
Math Myth Conjecture: If one restricts one's attention to the hardest cases, namely, graduates of top engineering schools such as MIT,  RPI,  Cal. Tech., Georgia Tech., etc., then the percent of such individuals holding engineering as opposed to management, financial or other positions, and using more than Excel and eighth grade level mathematics (arithmetic, a little bit of algebra, a little bit of statistics, and a little bit of programming) is less than 25% and possibly less than 10%.
  

This is a conjecture that desperately needs resolving with solid statistics and in-depth interviews.
 
Actually,  I'm already totally convinced of the veracity of the conjecture. This conviction is based upon the numerous scientists, engineers, and professors of science, engineering and math education that I've communicated with. In particular, Prof. Warren Seering, an engineering professor at MIT, who does surveys of their graduates, agreed with the conjecture. As did Prof. Julie Gainsburg, an ethnographer of mathematics in the workplace, whose done on-site work with structural engineers.

The following story also confirms the conjecture. Accenture, the former consulting part of the now defunct Arthur Anderson, was recruiting at UGA for math and computer science majors. I invited them-one professional recruiter and three consultants-to give their spiel to my class Math for Computer Science. After they finished, I asked the consultants: So, what mathematics do you actually use? They sheepishly responded: None. So, I asked them: What computer science do you actually use? Again the answer was: None. They were only interested in math or computer science majors as a convenient filter!
 
Acceptance of the conjecture should have revolutionary educational implications . In particular, it undermines the legitimacy of requiring higher mathematics of all students. Such mathematics is actually needed by only a minute fraction of the workforce.
  
There are two counter-arguments. The first is that higher mathematics is central to a serious higher education. This argument I agree with. Any Harvard undergraduate majoring in philosophy would certainly want to be able to understand Roger Penrose's book The Road to Reality. Unfortunately, that type of student is only a minute fraction of higher education. It is both unreasonable and unworkable to insist that all students get such training. Of course, such training should be available to all who desire it.
  
The second argument is the one I always hear around the mathematics department: mathematics helps you to think clearly. I have a very low opinion of this self-serving nonsense. In sports there is the concept of the specificity of skills: if you want to improve your racquetball game, don't practice squash! I believe the same holds true for intellectual skills. In any case, the case for transference of mathematical skills is unsettled. Moreover, mathematics is of little use in most problems of ordinary life. For example, mathematics could be of help in computing the costs of having children; but is useless in computing the benefits!
 
In conclusion, it would be wonderful to be able to get all students competent in Excel and arithmetic, and a little bit of algebra, statistics and programming. Higher mathematics should be offered and taken by those who need it, or want it; but never required of all students.


Comments and Sharing






COMMENTS (60 to date)
:-Daniel writes:

Fun fact: Elon Musk uses mostly Excel when he designs rockets.

David S writes:

As a fellow rocket engineer, even though we all do use Excel (I've done entire trajectory simulations in it - and you should see the rocket chemistry Excel page!), we also use log and exponential functions all the time. It is very helpful to be able to make rough estimates of those functions in your head. I also used very complex trig transformations when designing the skin of the vehicle I'm currently developing.

But I agree that we are the exceptions that prove the rule. I think Math is primarily useful for sorting out those that can handle complexity well. I do also believe it helps you think clearly, but I have no evidence beyond my own outlier experience.

andy writes:

Could you specify roughly what do you mean by 'higher mathematics'?

Thomas B writes:

I'm in your target school list. I no longer work in engineering, though. In the financial analysis part of my life, I occasionally use logs and exponentials - exponentials are very relevant in finance, of course. The concepts of derivatives (out to 2nd order derivatives) are commonly referenced by traders, on the occasions when I speak to them (as in, "it's going down but the 2nd derivative is improving"). I do some regressions, and here there is a need for some knowledge ("don't regress two exponentials", "beware of data mining") and some statistical common sense. Of course, there isn't much "common sense" in statistics, and it's not well taught in basic stats courses, either, as they tend to focus on numerous tests without building much intuition. In modeling customer bases and installed bases, I understand that the Excel models I build approximate integrals and, sometimes, convolutions - but someone who had never heard of either could still build those models. In my time leading a web systems team, I made good use of queuing theory, once (ever). I also modeled airflow over a hill using conformal mapping, once (that's grad school stuff).

I routinely tell people that everything I've ever needed to know about trigonometry, even as an engineer, you can learn in under an hour. Many people can grasp it thoroughly in 15 minutes.

One observation: advanced high school math tends to focus on problems that are tractable, with closed-form solutions. In the real world, problems are not tractable, or have no closed-form solutions. Numerical methods are required, and numerical methods, at their heart, are basic arithmetic. So, as with my customer base models, someone may be doing something that could be seen as "advanced math", or could be seen as "basic arithmetic".

What seems important, though, is fluency: I often find myself pouring out Excel work to conduct an analysis, but if someone should ask me what the plan is, I'd have to stop and think about it. I just "know where I'm going".

Richard Fulmer writes:

My Mechanical Engineering degree got me in the door. After I got hired, I used very little of what I'd learned in school. Instead, I picked up most of what I needed to know on the job.

Njnnja writes:

Your argument doesn't make sense: by focusing on the outcomes of the hardest cases (eg MIT grads) you then extrapolate to what "everyone" should do. The hardest cases (smartest people) don't do math because they rise up the ranks so quickly that they are managing people very quickly in their careers. The average person is going to be working for them in a "plumbing" job so it is important that they have technical skills. And even though the managers might not be coding on a daily basis, they need that background knowledge in order to supervise the work of others (which is why we make dark humor about the job prospects of English majors, even from great schools).

Further, I think we would be a much more statistically literate society if everyone had to take a calculus based probability class.

The plural of anecdote is not data, but from my own experience, if someone has some statistics, some programming (not hardcore, eg R or Python) and a little bit of systems knowledge, they can come out of graduate school at 25 and make well into 6 figures, more than a mid career engineer or actuary. And about 75% of the hires I have made and seen are immigrants because 10 years ago, Americans like you didn't want their kids to learn math and programming, and that didn't work out so well ("Would you like a tall or a venti?")

The jobs of the future are neither code monkey nor excel jockey, so looking at the actuarial profession in 2016 (which is pretty much the same as the actuarial profession 20 years ago) is not a useful example.

Chris Stucchio writes:

I'll go ahead and defend the use of math to make people clearer thinkers. By "clearer thinkers", I don't mean that math infuses into their brain improving general thinking.

I mean that many thoughts people have are simply nonsensical, and that nonsense evaporates when you try to put numbers/symbols to it.

For example, consider Henry Ford paying his workers enough to buy his cars. It's a quote I hear constantly parroted by innumerate newspaper columnists.

And for anyone who can do basic arithmetic, it's a quote that is immediately revealed as nonsense. If you pay your workers more money, profit goes down even if they spend every penny of their wages on your products. Just set up the accounting identities and this becomes apparent.

Math ensures your mistakes are consistent and makes your assumptions explicit. A lot of people could avoid many mistakes if they did more math.

Robert writes:

This is a really interesting essay, and it makes some good arguments. However I have some criticisms. I'm an actuary, and I have definitely used maths more advanced than "Excel and eighth grade level". I also have a PhD in maths (my thesis was on a topic within group theory).

I think the author conflates "is used by" and "is needed by". They are not the same, and in my experience there are actuaries who don't use as advanced mathematics as they need to for the problems they are tackling. A further related point is that I think the author is implicitly viewing maths as a tool that can be taken out of the tool box and used when needed, and then put back again. I don't think this is a good way to think about the issue. Thomas B said:

"I routinely tell people that everything I've ever needed to know about trigonometry, even as an engineer, you can learn in under an hour. Many people can grasp it thoroughly in 15 minutes."

I find this unbelievable. If anyone thinks they've grasped anything thoroughly after 15 mins (or even an hour) I can only guess that they've never understood anything thoroughly at all. I'm constantly relearning "basic" material and each time I develop a deeper more nuanced understanding of the topic, in particular this often involves seeing and understanding connections between different ideas that I hadn't understood before.

From this point of view the argument of learning higher mathematics becomes stronger. By learning more about the context within which problems originally arose and were studied, and about the more abstract theories that grew out of the original solutions you develop a deep understanding of the tools that do get used. You understand their strengths, limitations, and also ways in which they can be adapted and extended. You also become aware of different tools that aren't yet used, but could be useful.

A partial analogy would be with viewing art. Anyone can go into an art gallery and look at a picture or sculpture and appreciate it. They might even think that after 15 minutes they "understood" it. But their appreciation and understanding will be enriched by learning about the artist, the artist's influences, the historical context in which the art was produced and so on. This process of learning and understanding never reaches an end point.

I'm also unsure about the claim that maths skills are not transferable. Obviously the detailed knowledge of group theory that I built up while studying my PhD is not useful to me as an actuary. But the skill I did learn that is very useful is a more general ability to think abstractly, and more particularly developing a sense of the appropriate level of abstraction to think at. This later skill I had to learn while doing my PhD. This skill is definitely transferable, and I don't think it is learnt from studying non-mathematical subjects (except perhaps philosophy).

Ben H. writes:

I use higher mathematics routinely in my work as a scientist. I use complex statistical models like GLMs and GAMs; I program in R, C++, and other languages; I use trigonometry and occasionally calculus. And I use instincts developed during study of higher mathematics all the time – rates of change, equilibrium of dynamical systems, approximation and estimation and visualization. And yet my biggest weakness as a scientist is that my math skills aren't strong enough; I have been confronted by situations where I have needed to use FFTs, and Bessel functions, and partial differentials, and other such things, and my math chops have not been up to it, and that has been frustrating; I wish I had continued to take math for longer than I did (I mostly stopped after first-year calculus). No doubt I am an outlier, but I think my situation is still important and informative, as I'll explain below.

The objection I have to this essay is in the statement "Higher mathematics should be offered and taken by those who need it, or want it". How on earth do you know who that is? The author is advocating that children in general should not be bothered with progressing beyond "eighth grade level mathematics". So starting in ninth grade, math classes would essentially cease (or switch over to simply reviewing previously taught concepts), and only a tiny minority of students who "need or want it" would go further? How does a ninth grader know whether they are going to "need or want" higher mathematics as an adult? If they put off studying higher mathematics until they realize, as adults, that they "need or want" it, they will find it much, much harder to learn. They will be set back for the rest of their lives. This is my situation, in fact. After first-year calculus, I had no idea that I would "need or want" anything more, and so I stopped. And yet here I am, twenty-odd years later, both needing and wanting more, and regretting that I stopped. If most people stopped learning new math after eighth grade, I think we would find that a great many people would turn out to "need or want" training that they didn't receive; and that would be very difficult to fix after the fact.

And Bryan, I've heard you talk about how a larger population is better than a smaller population because you get proportionally more great minds – more Einsteins and Rothbards and whoever you might choose to cite. But the genius of such people is often not apparent in childhood. If all children who think (or whose parents think) they don't "need or want" to develop beyond the lowest common denominator are not pushed beyond an eighth-grade level, how many such great people would we lose? And how many would we lose who are not "great", but who are pretty darn good, and are essential for our society to perform at its current level? If the ideas in this essay were taken seriously, I would predict that our progress in mathematics, science, and engineering would cease, and indeed, that we would soon regress, losing knowledge we once had because we would then have too few minds capable of passing it onward. Too many people would never even imagine pursuing a career in math, science, or engineering, because nobody thought they would "need or want" to do so. The wasted potential would be staggering.

brad writes:

I did not graduate from the target list of schools (though I work with lots of folks who did) and not in engineering, but my experience working in the finance/risk management industry is very similar to Thomas B.

I would say I use math beyond an 8th grade level frequently. As Thomas says exponential are important and a conceptual understanding of derivatives, taylor series (for use in regression), integrals. Matrix algebra can be really important for Monte Carlo simulations and doing stuff in Matlab. Probability is obviously important and a general understanding of random distributions and their higher moments as well as an understanding that the expectation of a non linear function of random variable is not the same as the function of the expected value of the variable. I use regression analysis daily as well as many of its more modern adaptations. Also in statistics knowing understanding how to transform a uniform distribution to another distribution is very helpful.

I mainly use Excel, but I wish I knew more R, Matlab, C++/C#, or Python as I sometimes things get too big for Excel.


But also echoing what Thomas B. said. I pretty much never derive a function or take a derivative or figure out the closed form of an integral. Once you have fast computers those things are just too easy to do numerically. Also though I studied optimization most of the time it easy to use the built in solver in Excel.

I would say trig is useless for me, but just last week I had to use it. I had to do about an hour of Wikipedia reading to refresh my memory.

Just one data point from an admittedly math intensive job.


TMC writes:

I mostly agree with Bryan, as those above surely fit into his model as the top few percent who need higher mathematics. Most people will never need calc. Learning some higher math certainly has value though. As stated above, you get clarity of thought and while years later you may not be able to do the math, you know what is possible.

Talking to the Accenture folks may have been a bit misleading. Word and Powerpoint are the needed skills there.

OTOH, students don't understand a math course until they pass the following course. In other words, you don't understand arithmetic until you can pretend to do algebra. You don't understand algebra until you can pretend to do algebra.

A corollary: Nobody understands cutting-edge mathematics, not even the people discovering it.

Dave Anthony writes:

Boolean math! It is so important to programming, excel, database queries, business workflow logic.

It's really not even that difficult to learn but there are a lot of people that don't know it.

Scott Murphy writes:

Considering we are already at 25% with the framing of the question. I think the real number is as likely to be higher as lower.

First, people may cycle through a set of jobs which require math. You have to know how to do it to get them but you don't stay there. Those jobs may make up 10-25% of engineering jobs but most engineers may do them at some point. The jobs I am thinking of are also essential to everything going on at a firm.

Second, the ordering and categorization seem arbitrary. What is "a bit of programming" or "use excel". Excel has a non-linear optimization calculator! I guess if I use that I am not doing math? How would I even know how to use it or what I wanted without some sort of acquired training?


Why is statistics considered below trig? Most engineers I know don't learn much stat till college. I sure didn't.

What does "a bit of statistics" mean? Was it asked like that? Is "a bit" everything short of econometrics?

I know EE's who do a lot of digital circuit reduction. Digital design is quite mathy, but if you asked it in a framing like: "isn't it mostly algebra". Now it is packaged up under the preferred answer of the day.

I know a ME's who have fluid equations written all over their board that they manipulate to solve various problems in transport. It is true that they are doing a lot of algebra and would probably say as much. But there is a lot that is embedded in their training having to do with the way the quantities interact that would go well beyond what an "8th grade algebra" understanding of the same set of equations would produce. I could still see them answering a certain kind of question as though they did nothing but this algebra.

Another kind of mathematical maturity that would be subtle to tease out of a survey has to do with model limits. In transistor simulation there are zones of operation that have to be discerned. A good EE working in such a field may never write an equation more complicated than the linear transistor model. But possessing an understanding of concepts like sensitivity and non-linearity are still essential to its correct application.

Scott Gustafson writes:

MV=PQ

Also the % change in the money supply plus the % change in money velocity is equal to inflation plus economic growth.

Getting from the first to the second requires taking a derivative - fairly simple calculus.

Do economics students need to know why the second follows the first or should they merely take it on faith?

Correction: "You don't understand algebra until you can pretend to do algebra." should be "You don't understand algebra until you can pretend to do calculus."

brad writes:

"In sports there is the concept of the specificity of skills: if you want to improve your racquetball game, don't practice squash!"

But in sports almost everyone lifts weights as part of their training. And running/sprinting/jumping rope or other aerobic/endurance conditioning is common.

If you told someone "I am not sure what sport this person will ultimately play, but I want you to prepare him" a base of weight lifting, intense cardio, and endurance training would probably be what you would have them do.

If you tell a school, "I am not sure what this person's job will ultimately be but I want them to have good general quantitative and logical reasoning skills" upper level math is probably a good investment.

Scott Gustafson writes:

I think another way to look at this is the difference between education and training.

Training prepares you for situations you are expected to run into, providing you with the tools to deal with the situation.

Education gives you a broader background which helps you deal with unforeseen situations. It provides you with ways to approach new situations.

Your conjecture asserts the we need more training and less education.

At some level, everyone needs training. At more senior and especially management levels, we need more education.

Swami writes:

I totally agree with Bryan,

The vast majority of high school graduates should be spending their time learning the basics of 8th grade math proficiently, with a bit of added statistics (which is important in modern society). Some college graduates should pick up even more statistics, and other should pick up more geometry, advanced algebra and even calculus. These are specialty fields.

I look at the education system and see nothing but lock in on the old upper class finishing schools of Victorian boys. Nobody designing the system from scratch today could justify all the wasted time in advanced math for people who will never use it. Why not replace it with something useful like Latin or Womyn's studies? (Sorry, I couldn't resist)

Xenophon writes:

I’m a software engineer, with a Ph.D. in SWEng. from Carnegie Mellon. I rarely use calculus directly in my day-to-day work, but I use many of its concepts all the time. When I do use my calculus, it’s generally because I need to prove that my variation on one of the standard algorithms preserves its worst-case (or average-case) performance.

At various times in my career, I’ve needed to use matrix algebra, graph theory (TONS of graph theory!), plenty of trig, the ideas of the lambda-calculus, higher-order functions, various logical-calculi, numerical analysis, and probably a bunch more I’m forgetting at the moment. And every programmer uses Boolean Logic every time they write, read, or think about any program.

On the other hand, there are huge swaths of higher mathematics that have remained completely irrelevent to my career. And somehow, I’ve never needed even elementary statistics.

When I was an undergraduate, I had a free elective to choose. One alternative I considered was a course on Graph Theory. I couldn’t imaging a more useless or boring topic, so I took something else. But the joke was on me, as less than a year later I had a full-time job that was heavily based on graph theory (working on compiler back ends, especially the optimization parts). And I spent the next decade needing ever-more understanding of graph theory.

Xenophon writes:

I seem to recall reading a study on what knowledge students took away from their math classes in high-school and college. If I recall correctly, the answer was that folks had pretty good knowledge from all but their most-advanced class. Recollection from the final class was weak -- except for folks whose career required them to use the math from that final class on a frequent basis.

Sorry about the lack of a reference, but it should be discoverable if anyone cares enough.

Larry writes:

Math is weight lifting for the brain. Every athlete needs to be strong. Every ageing boomer needs to do resistance training to stay healthy.

The scary thing about math is how many people are either afraid of it or have so utterly failed to achieve numeracy that they don't know the difference between a million and a billion, much less understand statistics.

Roger Sweeny writes:

Further, I think we would be a much more statistically literate society if everyone had to take a calculus based probability class.

I'm sorry to be rude but that is a breathtakingly clueless statement. If there actually were such a requirement, 90% of the people taking the course would fail. Of the remaining 10%, at least half would be "math zombies"--able to pick up enough to pass with enough practice but not really understanding, and forgetting most of it after the course is over.

What you mean is, "I think we would be a much more statistically literate society if everyone UNDERSTOOD WHAT IS IN a calculus based probability class."

That is very, very, very, very, very different. You shouldn't feel too bad, though. Most everyone who talks about education makes the same sort of mistake.

Asha writes:

Another data point: I'm a healthcare public policy consultant. My job requires higher than 8th grade math skills.

Peter Gerdes writes:

As a mathematician and someone who has taught college mathematics I can't say I'm totally convinced learning mathematics isn't beneficial. I do believe it improves reasoning ability.

However, what we actually teach 95% of students (calculus, trigonometry, basic linear algebra) isn't math it's useless symbol manipulation. These problems can be completely handled by computers.

Worst all all not only do the things that we call mathematics encourage students to detest mathematics, quantitative thinking and instills behaviors that make that teaching actual mathematics very difficult.

As actual mathematics is about creative problem solving and figuring out how to solve problems you haven't seen before. I've had a great many students (when college calc courses try and teach a little theory or they take their first intro to real math class) simply unable to bring themselves to even try to solve a problem they didn't know an algorithm for and be totally psychologically unable to accept the idea math wasn't about 'solving' something.

I'd go even further and wouldn't even teach fractions to most children. Unfortunately, like most modern teaching, it's all about the signalling and smart students signal STEM proficiency.

James writes:

A few others have pointed this out but I'll pile on. It is impossible to anticipate what specific skills people will need in their future work and besides, it would be infeasible to give everyone an education customized to their future employment. Whatever schools teach, it probably won't be something that the students use later. So the right question is "What is the least useless subject that students can learn?" Math seems like a fairly good attempt at an answer.

Without disputing anything that anyone has said about the merits and uses of Math education, I disagree with an unstated presumption that lies behind much of what has been said, that strangers may usefully prescribe curricula for other people's children. The NEA/AFT/AFSCME cartel's K-12 tax-subsidized school system (the "public" school system) is the world's third-largest command economy, after China and Russia. Compulsory unpaid labor is slavery. Children work, unpaid, as window-dressing in a massive make-work program for dues-paying members of the NEA/AFT/AFSCME cartel.

Michael writes:

I wonder whether higher mathematics might be the new Latin. Even fifty years ago, educated persons knew some Latin (and probably French.) No one needed it day-to-day, but fluency demonstrated clear thinking and undergirded clear and effective communication. Perhaps related to the decline is a common complaint about today's graduates (especially engineers): poor writing and communication.

JayT writes:

I'm a programmer with a math degree, and I rarely use my math knowledge directly. However, I do feel like the way I approach problems is heavily influenced by that knowledge, and it has made me much more systematic about how I live my life.

That said, I think my biggest complaint is that if math isn't the focus, then what is? I personally found little to no value in any of the humanities courses I ever took.

darfferrara writes:

Arguing that people should be taught excel is where the essay went over the rails. Examples abound. Learn some python or R or Julia instead. The fact that Bryan uses excel to do his data analysis reduces my confidence in many of his positions.

David K. writes:

I have an undergraduate degree in physics from a decent (not top-tier) school and work in a management/finance position.

I agree with Professor Caplan: America wouldn't necessarily be better off with more people trained in higher math.

But I do believe the America/the world would be better off if more people were CAPABLE of being trained in higher math. That may just be a statement of the quality of our genetic stock; the world would be better if more people were more intelligent and high functioning.

I also agree that many people trained in higher math probably don't use it much, if at all. But I do think people with undergraduate degrees in math/engineering/compsci have something useful beyond some specific technical training.

Much of that usefulness may simply be what they are, not what they learned in college. For example, they're the kind of people capable of doing intellectually challenging work, and they're the kind of people who actually choose to do so.

I have an acquaintance with an econ undergrad degree and (non-econ) masters degree, both Ivy League. He once related a story about coming to an econ final hung-over and passing the test only because a friend shared his answers (i.e. helped him cheat). I was incredulous; he told me it wasn't a big deal: "all econ majors at XXX cheat".

I'm sure he was exaggerating, but I don't think a degree in some social science, Ivy League or otherwise, means much. It certainly means you chose not to do something more challenging given the opportunity. You may have recognized a college degree would be useful, but weren't interested in or capable of applying yourself very hard.

I don't even know if cheating in an advanced math class would be possible. I suspect "very very few math/engineering/comp-sci majors at XXX cheat."

I am biased toward hiring people with these types of degrees. I'm not particularly interested in their specific training. I'm filtering, as Professor Caplan says, for honest, hard-working, high-functioning candidates.

poorlando writes:

Bryan has clearly hit a nerve with this post. He should have known that he was barking up the wrong tree by espousing less math education to readers of a blog named ECONlog.

Aside from the standard math that we learn in high school, I think most people should have a basic foundation in probability. Looking back on my studies toward a PhD in a physical science, I think I would have been well served if I had taken more (read “any”) probability and statistics as an undergrad. In fact, I wish that high school had made us take calc in 11th grade and probability and statistics in 12th. More generally, I think that the world would be a better place if everyone had a better understanding of uncertainty, inference and causation — which underlies our personal and institutional expectations and decision making — through a better understanding of statistics. But for that to happen, the statistics curriculum needs to be torn down and rebuilt on a Bayesian foundation, with the menagerie of frequentist ad hocery reserved for a later appropriate time, if only so one can understand the language of the freqs and their inane, if not insidious, null hypothesis testing, p-hacking and otherwise strange and pathological way of understanding the world.

Glen Smith writes:

You cannot know who needs the higher math. The strategy I've seen is based on teaching it backwards which filters out most of your potential students, creates the math zombies who will likely fail the next class and helps insure that only the small percent of those who actually need it are those who continue on. By the way, after I finally understood primary reason to know basic calculus was (years after I first experienced it and hated it), it has become my most used form of math. Algebra is probably my least used.

Michael South writes:

My recommendation on this difficult question is that you institute a free market in education and let the market sort it out.

This is analogous to the calculation problem in the socialist commonwealth--there's absolutely no way that you can effectively centrally plan education. You don't even have a way of knowing what's best for one kid. There's no reason to move everyone through in lock-step; no reason to put people in with age cohorts instead of interest cohorts; no reason to force kids through classes that they will forget the content of immediately and resent you for forcing them through. We get all of that from the fact that we centrally plan education. We should stop that.

Robbo writes:

As a Maths graduate, having worked as a programmer, systems designer etc., and then on various management roles:

I have used lots of basic algebra, some statistics - which unfortunately my colleagues rarely understood very well - some queing theory, but little else from my higher maths toolbox.

However, I have found mathematical thinking, which I might describe as deep understanding of a few principles and the habit of taking them to the extreme, to be very useful at all levels of every job I have ever done.

In terms of education I would like to see everyone taught this mathematical thinking without having to learn it as a byproduct of studying geometry, calculus, vector algebra etc.

mathematical thinking is, as someone said above, as useful as the physical strength we train through weightlifting.

Graham Asher writes:

"Any Harvard undergraduate majoring in philosophy would certainly want to be able to understand Roger Penrose's book The Road to Reality."

Surely this is a huge exaggeration. "The Road to Reality" is 1000 pages long, and moves beyond even higher mathematics by about page 200. I agree that it would be nice if philosophy students knew about complex and transfinite numbers and had some idea of calculus, but to expect them to understand Riemannian manifolds, fibre bundles, tensors, spinors and the further reaches of mathematical physics is asking too much by almost an (uncountably) infinite amount.

GU writes:
In sports there is the concept of the specificity of skills: if you want to improve your racquetball game, don't practice squash!

I don't know much math, but I have a lot of experience with athletics. While it is true that sport specific training is necessary—and the most important thing an athlete can do—it is not true that other training doesn't produce robust positive returns to your sports performance.

Specifically, getting stronger is usually the easiest way to improve performance. There are diminishing returns of course—a triathlete might see them once he can squat 300 lbs, a shot putter may want to get his squat to at least 550 lbs, etc. But the "carryover" of generalized strength training to specific sports is quite significant.

Does that mean learning math helps you think more critically or analytically in general? I have no idea. But if sports is an apt analogy, then it probably does help somewhat.

David writes:

Edward's argument that math fluency doesn't necessarily lead to clear thinking is profoundly and humorously reinforced here by the number of mathematically literate commenters who think "I use math!" is a counterargument to the proposition that most people don't.

Robert Arvanitis writes:

Oliver Wendell Holmes said there are A,B,C minds and X,Y,Z, minds ("Autocrat of the Breakfast Table"
By that he meant the former require concrete examples while the latter can handle abstraction.

To be effective voters, people must understand enough math to grasp blatant frauds.
Jonathan Gruber chortled on gaming CBO scoring of Obamacare: seven years of costs against 10 years of revenue.
Joe Biden bloviates we must spend to get rich.
Price controls - like renumbering the thermometer and claiming the fever is gone.

You needn't be able to handle Navier-Stokes in order to grasp obvious algebraic absurdities.

Njnnja writes:

@David -

BC (actually DE): Going to the gym is a waste of time! When are you ever going to bench press 200 lbs in real life? People ought to sit at home watching TV
Commenter: I go to the gym, and I am healthy, happy, and will live longer than people who don't. Therefore other people should also go to the gym.
David: @commenter - But most people do sit around and watch tv. Therefore they ought to sit at home watching tv

(See is-ought fallacy).

Steve J writes:

I don't get where Bryan is going with the "most people don't need to be educated" reasoning. Maybe he is trying to postpone automation by having more low skill labor available? He doesn't even propose an alternative filter for finding the smart people. The education filter may be inefficient but it is effective. High performing students at the schools he mentions are either very smart or very hard workers - I don't remember meeting any exceptions.

David writes:

@Njnnja

For your analogy to be relevant, it would need to be a given that higher-level math skills, like exercise, benefit all. But isn't that precisely the debate?

Note also that it is difficult to be guilty of the is-ought fallacy when one has yet to offer an opinion on what ought to be. I simply pointed out that Edward's Math Myth Conjecture is not refuted by any individual's use of or experience with higher math.

(Steve): "I don't get where Bryan is going with the "most people don't need to be educated" reasoning."
Who gets to define "education"?
It does not take 12 years at $12,000 per pupil-year to teach a normal child to read and compute. Most vocational training happens more effectively on the job than in a classroom. State (government, generally) provision of History, Economics, and Civics instruction is a threat to democracy, just as State operation of newspapers and broadcast news media would be *and are, in totalitarian countries like Cuba and North Korea).

Jeffrey Eldred writes:

A study on Norwegian college admissions finds that most individuals are better off in their chosen field of study than their next best choice, even when that next best choice is generally a more lucrative field:
https://www.bc.edu/content/dam/files/schools/cas_sites/economics/pdf/Seminars/SemF2014/Mogstad.pdf
My interpretation of the findings is that college skills are transferable. Its far more important to motivate a student to work hard in any field at all than it is important what that field of study actually is.

The technical knowledge that employers require can only be obtained by being an employee. The next best thing is someone trained in rapidly acquiring new technical information.

In my own experience as a math major, I found myself puzzled by the organization of undergraduate math. At that level, every branch of mathematics is very different and almost every math class starts from scratch in some sense. Yet every math class I took became easier than the last.

I wasn't drawing on my prior math knowledge, rather I was getting better at learning math. This, the ability to absorb new technical information and do something sensible with it, is the skill employers are after.

The same is true of writing. Employers are much more likely to need technical writers than academic writers, yet academic writing is precisely the training employers rely on for their technical writers.

Steve J writes:

@Malcom - I agree we do not need as much education as we get. But as Bryan admits we are using education mainly as a signal for determining who are the intelligent, hard working people in society. I agree with him again that it is an inefficient filter but I am not aware of a better one. If your company has well functioning hiring processes you may be able to find the great candidates that did not make it through the education filter but it is a LOT easier to find great candidates from MIT.

The part people do not seem to admit is just how hard it is to separate the wheat from the chaff. Education is somewhat like sports in that the playing field is supposed to be level and we can tell who the best players are. Outside of sports and to a lesser extent education the amount of luck involved in most success stories blurs the amount of talent involved. You don't luck your way through the engineering program at a top university. The people who make it through know what they are doing.

Njnnja writes:

@David

I don't think commenters are using their anecdotal experience to refute the math myth so much as they are refuting the "should" in the conclusion. At least that was my original point.

Do not confuse attendance at school with education. Do not suppose that an institution which someone calls a "school" provides education. On-the-job training provides education. If P.E. class provides education then so does recreational rock climbing.

Becker (__Human Capital__) defines "school" as an institution whose principal product is education. A couple of points:
1. A manufacturing enterprise or extractive enterprise might produce more education, per new employee per year than will one of the NEA/AFT/AFSCME cartel's so-called "schools".
2. Some of the cartel's school generate employment for dues-paying members of the cartel and otherwise produce only angry unskilled illiterates after 12 years.
3. Operators in an unsubsidized competitive market in education services would face strong incentives to make self-paced subject-matter delivery more palatable. I taught Set Theory and logical notation to a third grader who skipped high school and college and got his MS (Math) before he turned 19.
School is for dummies -- Bobby Fischer

jb writes:

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Thomas Sewell writes:

Peter Gerdes,

Where are some resources/classes/books, etc... someone could use to learn primarily (in your terms) actual mathematics rather than symbol manipulation?

I'd be very interested in finding a way to be able to have someone learn the one without having to wade through the 90% formula and step memorization crap which seems to pass for math education many places these days.

Reading Euclid and Newton directly only takes someone so far.

Thanks.

Brian writes:

Is it possible that the 10% using these skills are he only ones doing real engineering? After all, most premeds don't become doctors. Most law school grads don't do trials. And most engineering titles are held by people who do program management, requirements review, and other paratechnical jobs.

I work in computer security. I have a book and some Wikipedia pages open right now on queuing theory, so exponentials, approximations, and calculus are right here with me. Last month was crypto, and through it all we apply ideas like Shannon information theory---which I think you can't understand with only eighth grade math.

ed writes:

Related:

https://www.washingtonpost.com/news/wonk/wp/2016/08/26/an-alarming-number-of-scientific-papers-contain-excel-errors/

JB writes:

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Evan Zamir writes:

It's no wonder most people have no clue how to invest their money. If you don't know what an exponential is, you probably can't understand the idea of compounding interest well enough to see how important it is.

Adam Scott writes:

Higher math is for mathematicians. Mathematicians are amazing. Therefore higher math is for amazing.

QED

Bruce Dale Wise writes:

I really enjoyed David Edwards The Math Myth, and all of the following commentary. I am a teacher concluding his career. My main field is literature and writing; but as the years rolled by I found myself teaching more math classes, and have taught at the high school level, in addition to a basic computer programming class, Algebra 1, Geometry, Algebra 2, Trigonometry, Calculus, Advanced Math, and Applied Mathematics. As an old-school teacher, I have this certificate which allows me to teach anything K-12.

Anyway, the math situation is complex.

1. There is more than just one math myth; but the one Edwards refers to is that the future of the American economy is dependent upon we masses having higher math skills. Competition with the Russian in the 1960s, with the Japanese and the Germans in the 1980s, and with the Chinese and the Indians didn't hurt us then and won't hurt us now. Besides, we can learn from others. By way of an anecdote, the top two math students I have had over the years were both Russian foreign exchange students, who told me that they took up to three math classes a year. It is true that when I was a student in the 1960s, and we were taught the New Math (basically cullings from Group Theory and Set Theory), it seemed very odd to me at the time; but as the years have progressed I am thankful for having been put through that. That little bit of middle school math actually helped me to see later that math was becoming more abstract historically, and that its foundations were even more complex than the greatest mathematicians of the early 20th century realized (Peano, Frege, Russell, Hilbert, et. al.).

2. It is true also, as some have suggested, that calculus is used as a filter for some jobs: architect, pharmacist, chemist, computer programmer, accountant, geologist, dentist, engineer, physicist, business manager, doctor, etc. Yet it is also true that many jobs do not use much more than Excel and eighth-grade math on the job. So why is it globally required? There are many reasons. I'll name just a few. Math is a repository for logic, measurement, practice with algorithms, problem solving, working with symmetry, understanding complexity, practical computation, clear thinking (though maybe not for Professor Edwards), etc. And by the way, in calculus one practices all previous math learned (arithemetic, algebra, geometry, statistics, etc.), in addition to limits, derivatives, and integrals.

3. How much math is required for each individual depends on what the individual is going to do in life; but from my experience as a high school teacher, I have found a large number of students are unsure of what they plan to do in life. And even so, the very jobs themselves are changing. Yet since, there is not one size of math that fits all, I always check with high school students in my calculus classes to understand why they are there. Some do not know what they plan to do in life, and take calculus to keep their opportunities open. But others do know, and I check with them. A student who wants to be a veterinarian needs it for entry into vet programs across the nation, but high school algebra (not an 8th grade level) will suffice for a registered nurse. I make sure the students realize this. This is where I concur with Edwards; one math size does not fit all. I would say further, not higher, mathematics should never be required of all students. Still, I'll never forget one girl who took two years of calculus simply because she wanted to help her future children when they went to school. She is now a mother of three.

4. I do think Professor Edwards is on the wrong side of history. I doubt whether his self-convinced conjecture will be taken as legitimate at all. First off, people are drawn to the professions that make the most money, economics, medicine, technology, and science; they want more mathematics. With about 7,400,000,000 people on the planet, does he really think that a majority of them will be satisfied to have an eighth grade level of math and a little programming? Would he? I sure as hell wouldn't. We are mathematical beings, from our geometrical bodies and minds to the universe we find ourselves in, and I daresay we wish we had even more refined mathematics to achieve an even greater vision of reality than that which we possess at present. I am thankful for Professor Edwards conjecture; it is a nice reminder of overzealous views on education; yet, I must admit as to thinking his conjecture itself is a bit of myth.

Tyler Wells writes:

I am a humble cost accountant with a State U education and my job would probably satisfy the “eighth grade math and Excel” criteria well. I have never overtly used the calculus I learned, for example.
That said, I have never said “I wish that I knew less math” or even that I hadn’t “wasted” my time taking math. To the contrary, I often lament that my employees and co-workers labor over the ideas, including math, needed to complete their jobs. To be honest, I was the same way myself when I entered the workforce.
While I would agree that the vast majority of Americans don’t need higher math to flourish in their professional and personal lives; I would argue that they need a deeper math in order to achieve those objectives. I’m old school, over 25 years out of high school, and my high school and college math taught me a great deal about calculation but very little about application. However, judging from my nephew’s math books I don’t see that there has been much change. I would argue that there should be more math in schools, not less, and that this math should be focused on solving the everyday problems that students will face as well as potential career issues.
As an example, one of my own pet peeves is how many people misunderstand interest rates and the time value of money. How much are your student loans going to cost you? How much of a pay premium will that psych major earn you? Every high school student should be required (and able) to calculate the break-even point between entering directly into the workforce or studying for four or more years at a university. They could then be asked, “What if you changed to a different major, how would that affect your calculation” and so on.
Tons of problems that require assumptions, estimates, and inquiry by the students should be asked. In my high school math I was given all of the information necessary to complete word problems. Well, in my job, I rarely have all the information directly provided and I have to do research and to ask lots of questions, some of them uncomfortable. I am often not 100% sure that my assumptions are correct, that my information is complete, and that I am communicating the information fully and comprehensibly to others.
In short, even if professor Edwards is correct, I hardly see that what logically must follow is that the 90% of non-math elite, myself among them, stop taking math after eighth grade. Instead, I would argue for more math but with a different approach for these students.

David Otey writes:

I am a teacher of 22 years experience, teaching grade school and high school special education. I have enjoyed the previous comments from financiers and rocket science engineers which proves to me that the higher math needed for the higher level jobs can wait to be taught in the University arena where the real job preparation is supposed to be taking place. My beef with the education industry is wasting out students time from 7th grade on with both math and language courses. Better subjects would be money management, investment strategies, entrepreneur possibilities with students discovering their talents with teacher's help. Real job training CAN begin in Junior and Senior high school for lots of ''real''jobs: mechanics, construction, CNA, infrastructure jobs and more. The majority of U.S. high school students graduate with only two choices--what they actually QUALIFY for after 12 years of U.S. education--fast food joints and super store clerks, or sign up to start college where the first two years waste their time with repeating most of what they had in high school only at a faster pace with extra depth of useless material. Shame on this country for maintaining the antique industrial education system.

Plucky writes:

I'm coming late to this, so I hope I'm not screaming into the void here. A few thoughts

-At the most basic level, the basic thesis is correct, that the actual mathematical abilities most jobs require are not very advanced, and in particular the practical usefulness of trig and calculus is very rare.

However, I strongly disagree that math is over-taught (although I do think it could/should be taught differently), for a few reasons

- In professional settings, math skills are only useful when the mathematics is completely mastered. If you're only 93% certain that the person will get an interest rate question right, it's simply not responsible to rely on their financial analysis. 93% will get you an A in high school but will result in intolerably expensive mistakes in real life. In my experience, true mastery of math skills exists about 2 rungs down the ladder from the hardest math you've gotten a 'B' in. In order to get mastery of lower-level skills, you need to push people past them mathematically

- Until you get to 200's and 300's undergraduate level, math tends to be sequential and cumulative, which means that if you ever encounter a mathematical problem you can't deal with, you can't just go to the wolfram website and expect to teach yourself how to solve it in an afternoon. Odds are you might need months worth of study to get to where you need to be, which is generally not OK in professional settings.

- The payoff structure of a lot of math skills is like the ability to change a flat tire. You may only need the skill 2 or 3 times in your life, but when you do need it, you're really, really glad you have it.

- Related, lack of math in HS/college can effectively close off entire courses of study and career paths. If you decide in the spring semester of your Freshman year that you want to change majors and become an engineer, but haven't touched math since pre-calc your junior year of HS, you might be SOL. It might take you six months just to get to the point that you can start taking engineering classes, and plenty of people will say 'nope, not worth it' at that point. Teaching people "too much" math is in part making sure people actually have the opportunity to go into mathematically demanding fields when they are old enough to make that choice.

- Much more important than math "skills" is math "sense". Practically all calculation is handled by computers nowadays, and the real ability the human needs is the ability to spot errors and things that don't make sense, i.e. the ability to look at a result and immediately recognize "That can't possibly be right, it's off by an order of magnitude" and then be able to figure out where the problem is coming from. In general, people only acquire "math sense" when they've spent a lot of time doing math. I'd analogize it to language immersion. If people could figure out how to teach "math sense" without making people to harder math, then great, but I'm not sure how that would be done.

- Using math and computer science as a filter is not necessarily all that's going on. In my experience there is a substantial treatment effect to education in math and computer science, especially when it comes to rigor of thought. It's not just about filtering ability but actually developing it. The act of having to solve hard logical problems in CS or of having to make mathematical proofs changes the way you think in ways which are commercially valuable. Posters above have analogized it to strength training for athletics, and that's a reasonable comparison. It is probably not strictly necessary that math or CS be used to develop that rigor of thought, but is sure is an indictment of the humanities that companies that need to recruit it go for majors with limited direct relevance.

Fred Anderson writes:

I am now retired after 37 years of university undergraduate teaching. Before that, I was a marketing researcher for a decade.

I will never forget one experience from those marketing research years. We had done some kind of depth analysis of a large survey -- complete with hairy statistics -- and presented the case to senior management. Upon leaving the conference room, my immediate superior chewed me out: "Don't you ever do that again! You scared the He*ll out of them. Most of them didn't understand two sentences in a row of what you said. You've probably killed the project. Few of these men are going to gamble their careers on a multi-million dollar undertaking that they find completely incomprehensible."

He was right. Business management is a cooperative endeavor, and -- to succeed -- it requires buy-in from all the people who are going to have to get it done. We have a lowest-common-denominator problem here. Many of the people you must bring along are not familiar -- perhaps were never familiar -- with much math beyond percentages and averages.

Those of us doing the analyses need to know the math so that we can be reasonably confident we got it right. But in public presentation it may be better to attribute the conclusions to "the computer".

Xiang writes:

The problem with the American mass has never ever been the lack of "advanced math" or whatsoever. They're missing the point. It's the tremendous gap between elite education and "common" education, as well as the lack of very basic scientific common sense among the population. It's not required for people to possess outstanding advanced skills like a PhD, but when many get some of the most basic facts wrong, and even believe the earth is 4000 years old for example, then there's a massive problem.

Of course I know it's the elites among the upper echelons of the society who are more than happy to see and maintain such a situation, and unfortunately this article might well be another addition, a so-called academic/think-tank publication that serves their agenda. It can't get more obvious at the end of the article: "leave elite education to those who 'need' it! Keep the mass ignorant!" Yeah, sure, so that the children of the elites always stay powerful and the mass keep remaining ignorant. It doesn't matter for the massive power wielded by the US, the state terrorism employed by Uncle Sam, but it matters, a lot, for genuine empowerment of the people and true democracy, which people, potentially including the author here, doubtlessly want to stop at all costs.

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